Difference between revisions of "2021 AMC 12A Problems"
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==Problem 12== | ==Problem 12== | ||
− | + | All the roots of the polynomial <math>z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16</math> are positive integers, possibly repeated. What is the value of <math>B</math>? | |
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+ | <math>\textbf{(A) }-88 \qquad \textbf{(B) }-80 \qquad \textbf{(C) }-64 \qquad \textbf{(D) }-41\qquad \textbf{(E) }-40</math> | ||
[[2021 AMC 12A Problems/Problem 12|Solution]] | [[2021 AMC 12A Problems/Problem 12|Solution]] |
Revision as of 14:37, 11 February 2021
2021 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
What is the value of
Problem 2
Under what conditions is true, where and are real numbers?
It is never true. It is true if and only if . It is true if and only if . It is true if and only if and . It is always true.
Problem 3
The sum of two natural numbers is . One of the two numbers is divisible by . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
Problem 4
Tom has a collection of snakes, of which are purple and of which are happy. He observes that all of his happy snakes can add, none of his purple snakes can subtract, and all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Problem 5
When a student multiplied the number by the repeating decimalwhere and are digits, he did not notice the notation and just multiplied times . Later he found that his answer is less than the correct answer. What is the -digit number
Problem 6
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is . When black cards are added to the deck, the probability of choosing red becomes . How many cards were in the deck originally?
Problem 7
What is the least possible value of for all real numbers and
Problem 8
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 9
Which of the following is equilvalent to
Problem 10
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 11
A laser is placed at the point . The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the -axis, then hit and bounce off the -axis, then hit the point . What is the total distance the beam will travel along this path?
Problem 12
All the roots of the polynomial are positive integers, possibly repeated. What is the value of ?
Problem 13
Of the following complex numbers , which one has the property that has the greatest real part?
Problem 14
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 15
A choir director must select a group of singers from among his tenors and basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of , and the group must have at least one singer. Let be the number of groups that could be selected. What is the remainder when is divided by ?
Problem 16
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 17
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 18
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 19
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 20
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 21
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 22
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 23
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 24
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
Problem 25
These problems will not be posted until the 2021 AMC12A is released on Thursday, February 4, 2021.
See also
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2020 AMC 12B Problems |
Followed by 2021 AMC 12B Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.